#72321161

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂⁴	#442
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1349 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2)(3,4)(5,6,9,12,10,13,17,14)(7,11,16,18,15,19,20,8)(21,22,25,32,28,34,29,46)(23,35,37,50,40,48,41,52)(24,33,26,43,42,31,27,45)(30,36,47,38,49,44,51,39);(2,4)(5,7)(6,11)(8,14)(9,16)(10,15)(12,18)(13,19)(17,20)(21,23,28,40)(22,30,34,49)(24,39,42,38)(25,41,29,37)(26,44,27,36)(31,50,33,52)(32,51,46,47)(35,43,48,45);(2,4)(5,8,17,19,10,18,9,11)(6,7,14,20,13,15,12,16)(21,24,29,27,28,42,25,26)(22,31,46,43,34,33,32,45)(23,36,41,39,40,44,37,38)(30,35,51,52,49,48,47,50);(7,15)(8,18)(11,19)(16,20)

Polycyclic presentation:

2^10|a^2=e,b^2=j,c^2=eh,e^2=h,f^2=i,g^2=i,i^2=j,b^a=bf,c^a=cfgij,c^b=ci,d^b=dhj,d^c=dhj,e^b=ej,f^a=fi,f^b=fij,f^c=fj,g^a=gi,g^b=gij,g^c=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321249, #72321337, #72321785
maximal normal subgroup of	#72321249, #72321337, #72321785

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 3, F.5*F.8);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);